I vaguely remember that the gate is referred to kara kapi or siah kapi in Turkish.
Mesih Dirildi* Al Masih Qaam
Gerçekten Dirildi! Haqqan Qaam
I have not heard that kind of a designation, but only remember that a nationalist officer called it the gate of spite/grudge.
An interesting detail from political history: The current prime minister went to visit the EP after his first victory in politics as the mayor of Constantinople in 1990s and asked His All Holiness to open this specific gate as a sign of the EP's willingness for peace & forgiveness. Of course, he was turned down.
From time to time some conservative and pro-Ottoman newspapers remind their readers of this tragedy, but praise the slaughter of the Patriarch, whom they consider a traitor.